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DOI: 10.1039/C6CP04252J
(Paper)
Phys. Chem. Chem. Phys., 2016, Advance Article

Marie Humbert-Droz*^{a},
Claude Piguet^{b} and
Tomasz A. Wesolowski^{c}
^{a}University of Geneva, Geneva, Switzerland. E-mail: Marie.Humbert-Droz@unige.ch
^{b}Sciences II, Department of Inorganic Analytical Chemistry, University of Geneva, Geneva 4, Switzerland
^{c}University of Geneva, Physical Chemistry, 30, quai Ernest-Ansermet, Geneve, Switzerland. E-mail: Tomasz.Wesolowski@unige.ch

Received
17th June 2016
, Accepted 16th September 2016

First published on the web 16th September 2016

Terpyridine derivatives are of great interest due to their unique photophysical properties when used as antennas in metallic complexes. Several experimental and theoretical studies indicate strong charge-transfer character of the lowest electronic excited state, which could be exploited for predicting fluorescence quantum yields from the magnitude of the charge separation induced by electronic transitions. Focusing on substituted 4′-phenyl-2,2′:6′2′′-terpyridyl, we report on two measures of the charge separation obtained from high-level calculations in ground and excited states (length of the change of the dipole moment and the electron–hole distance). Our refined model confirms that the fluorescence quantum yield shows a global S-shape dependence on the magnitude of the charge separation, which can be quantified either by the change in dipole moments between the ground and excited states or by the associated charge–hole distances. This approach provides a remarkable tool for the molecular design of a fluorescent polyaromatic antenna.

Fig. 1 Structures of π-conjugated hydrocarbons used in ref. 1. |

Fig. 2 (a) Potential energy diagram showing light absorption (hν_{exc},k_{exc}), fluorescence emission (hν_{em},k_{r}) and radiationless (k_{nr}) processes for the excitation of conjugated aromatic molecules. (b) Simple electron-transfer model proposed by Yamaguchi et al.^{1} illustrating the π-conjugation length (A_{π}) in the S_{1} state of a π conjugated molecule. |

In order to model the fluorescence quantum yield Φ_{F} = k_{r}/(k_{r} + k_{nr}), both radiative k_{r} and non-radiative de-excitation processes should be at hand (left part of Fig. 2). Obviously, the S_{0} → S_{1} transition is expected, and indeed does, modify the electronic structure of the aromatic molecule,^{2} a process assigned by Yamaguchi et al. to the formation of a permanent dipole in the S_{1} state represented by the so-called π conjugation length (see Fig. 2).^{1}

With this hypothesis in mind, the radiative emission rate constant k_{r} is given by Einstein's equation where A_{1,0} is the Einstein transition rate constant for spontaneous emission (in s^{−1}).^{3–5}

k_{r} = A_{1,0}
| (1) |

Since the energy gap between the S_{1} and S_{0} levels in polyaromatic molecules is usually larger than 20000 cm^{−1} (Table S1 in the ESI†), any deactivation due to coupling with high-energy phonons can be neglected.^{6} The non-radiative rate constant (k_{nr}, the sum of intersystem crossing and internal conversion)^{7–9} was thus assigned by Yamaguchi et al.^{1,10} to the relaxation of the induced dipole resulting from the intramolecular electron transfer. The constant k_{nr} can then be modelled by means of the Hush equation:

k_{nr} ≡ k_{ET} = k_{0}e^{−βR}
| (2) |

Taking βR = A_{π} as a measure of the π conjugation length of the aromatic system in the S_{1} state, its limiting values k_{0} for A_{π} = 0 were assigned to Einstein transition probability for stimulated emission B_{1,0} (in [m^{3} J^{−1} s^{−2}]) induced by the spectral energy density ρ(ν) (in [J s m^{−3}]) produced by an isotropic radiation field at the frequency ν of the emission band (h is Planck's constant, k_{B} is Boltzmann's constant, c is the vacuum speed of light and T is the absolute temperature).^{1,3–5} Let us stress here that the original derivation^{1} neglects the contribution of the spectral energy density. This approximation can be lifted in a straightforward manner with the introduction of the Boltzmann factor exp(hν/k_{B}T) measuring the effect of the energy band gap on the non-radiative relaxation process:

(3) |

From eqn (1) and (3) it follows that:

(4) |

(5) |

Note the similarity of eqn (5) to eqn (10) from ref. 1, which can be cast in the form:

A_{π} − A^{ref}_{π} = ln(k_{r}/k_{nr})
| (6) |

Eqn (6) mirrors eqn (5) of the present work if all members of the family of aromatic hydrocarbons under investigation possess the same energy gap hν_{ref} separating the S_{1} and S_{0} states (then A^{ref}_{π} = −hν_{ref}/k_{B}T).

Since the fluorescence quantum yield Φ_{F} is defined as:

(7) |

(8) |

A_{π} can thus be easily computed for a family of compounds as soon as their quantum yields and emission energy frequencies are available (Table S1, column 10 in the ESI†). The plot of A_{π} computed for compounds **1–3**, **P0–P3** and **C1–C4** against their longest molecular axis (Fig. 3) indeed suggests the existence of an intriguing correlation between the experimental spectroscopic A_{π} parameter and some molecular dimensions attributed to the π-conjugation length.^{1}

Fig. 3 Plot of A_{π} against longest molecular lengths taken from ref. 1 in the ground state for conjugated (black diamonds) and fused (red squares) aromatic hydrocarbons (built using the data reported in Table S1, ESI†).^{1} Assuming a 10% uncertainty for the measured Φ_{F}, the size of the resulting error bars on A_{π} is smaller than the size of the symbols. The error bars are thus not shown. |

Pushing further this strategy, the next step consists in the identification of one or several physically-relevant electronic properties which could be used as reliable estimations of A_{π} parameters. The obvious choice suggested by Fig. 2b^{1} is the length of the difference between the dipole moments (|Δ|) in states S_{1} and S_{0}. If |Δ| is expressed in [e Å], its numerical value corresponds to the charge separation in Å. Moreover, the vector Δ provides the information about the direction of the charge transfer. Unfortunately, this simple measure of charge separation is not applicable for centrosymmetric molecules **1–3**, **P0–P3** and **C1–C4** gathered in Fig. 1.^{1} In such a case, the first non-vanishing change in the electric moment has a quadrupole character, and the interpretation of the change in the quadrupole moment as the displacement of electric charge is not unique.

The alternative charge–hole separation distance (d_{he}), evaluated for the one-particle transition density matrix,^{2} may independently support the change in dipole moments, but it also vanishes for centrosymmetrical molecules. In order to remove symmetry limitations in the interpretation of A_{π}, we decide to compute |Δ| and d_{he} for a series of non-centrosymmetric 4′-phenly-2,2′:6′,2′′-terpyridyl compounds **1a–1g** (Fig. 4), the photo-physical properties of which were recently reported which allow us to determine the experimental A_{π} parameter using eqn (5) (see Table 1).^{12} It is worth reminding here that terpyridine derivatives are of great interest due to their unique photo-physical properties if used as sensitizers for transition metals. Several experimental and theoretical studies indicate strong charge-transfer character of the lowest electronic excited state^{12–14} and the presence of nitrogen atoms in the aromatic molecules **1a–1g** (compared with the pure hydrocarbons gathered in Fig. 1) is known to reinforce the spin–orbit coupling constant. Intersystem crossing (ISC) processes then become the dominant contributions to the non-radiative deexcitation rate constants of the S_{1} states, which optimizes light downshifting in luminescent metallic complexes.^{7–9,15}

Cmpd | Φ_{F}^{a} |
λ_{em} (nm)^{a} |
ν_{em} (s^{a}^{−1}) |
logε^{a} |
λ_{abs} (nm)^{a} |
τ (ns)^{a} |
k_{r} (s^{a}^{−1}) |
k_{nr} (s^{a}^{−1}) |
A_{π} eqn (5) |
hν/k_{B}T |
|Δ| (e Å) | d_{ch} (Å) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

a Data taken from ref. 12. | ||||||||||||

1a |
0.18 | 360 | 8.33 × 10^{14} |
4.89 | 291 | 2.14 | 8.41 × 10^{7} |
3.83 × 10^{8} |
−135.7 | 134.2 | 1.51 | 5.05 |

1b |
0.16 | 358 | 8.37 × 10^{14} |
4.86 | 287 | 1.77 | 9.04 × 10^{7} |
4.75 × 10^{8} |
−136.6 | 134.9 | 1.12 | 4.64 |

1c |
0.17 | 359 | 8.35 × 10^{14} |
4.87 | 289 | 1.48 | 1.15 × 10^{8} |
5.61 × 10^{8} |
−136.1 | 134.5 | 0.43 | 4.12 |

1d |
0.50 | 403 | 7.44 × 10^{14} |
4.71 | 298 | 2.15 | 2.33 × 10^{8} |
2.33 × 10^{8} |
−119.8 | 119.8 | 2.58 | 5.43 |

1e |
0.73 | 447 | 6.71 × 10^{14} |
4.52 | 296 | 1.51 | 4.83 × 10^{8} |
1.79 × 10^{8} |
−107.1 | 108.0 | 4.05 | 6.38 |

1f |
0.92 | 409 | 7.33 × 10^{14} |
4.56 | 327 | 0.77 | 1.19 × 10^{9} |
1.04 × 10^{8} |
−115.6 | 118.1 | 1.50 | 5.32 |

1g |
0.60 | 489 | 6.13 × 10^{14} |
4.60 | 284 | 2.68 | 8.41 × 10^{8} |
1.79 × 10^{8} |
−98.4 | 98.8 | 3.67 | 6.30 |

For each considered compound, the charge-separation magnitudes (|Δ| and d_{he}) were evaluated in the following procedure. (A) The geometry of the isolated species was optimized at the ground state. (B) Electronic excited states were obtained. The lowest state with a high oscillator strength (the one corresponding to the lowest maxima of the absorption band) was identified. It was usually the lowest excitation. The calculated vertical excitation energy was compared to the experimental absorption band maximum in order to validate the used methodology. (C) The charge-separation magnitudes |Δ| and d_{he}, which were used as the first-principles counterparts of the empirical parameter A_{π} in subsequent analyses, were evaluated at the ground-state geometry for the state identified in step B. This simple procedure providing numerical values of the charge-separation magnitudes is obviously not general. It neglects the fact that fluorescence occurs at excited state geometry, the solvent effects on both the geometry and the electronic structure, and the fact that if the state with the highest oscillator strength appears not to be the lowest excitation (three out of seven investigated compounds) violating thus Kasha's rule. Last but not least, the procedure is not applicable if both ground and excited state geometries remain centrosymmetric. The compounds chosen for the present analysis are not centrosymmetric and are rigid. Their geometry is not expected to be significantly affected by either the solvent or the excitation. The effect of the solvent and the relaxation of geometry upon excitation might change the order of states nevertheless. Without explicitly modelling these effects, we have chosen to evaluate the charge separation magnitudes for the same state assigned to the lowest maximum of the absorption band assuming that it becomes the emitting state in the excited state geometry and in the presence of the solvent. We underline that the primary objective of the present work is to verify if there exists a relation at all between the empirical parameter A_{π} and the quantum mechanical observable measuring the charge-separation magnitude.

Interpreting the character of the transition looking at the orbitals (Fig. S15–S21 in the ESI†), our CC2 or ADC(2) calculations give transitions of ICT characters for compounds **1c**, **1e**, and **1g** (strong) and a weak ICT character for compound **1f**. In contrast, compounds **1a**, **1b** and **1d** show an excitation delocalized on the π system. Analyzing the transition in terms of electron–hole densities, using the one particle transition density matrix,^{2,25} gives a more precise picture on the nature of the excitation. Fig. 5 shows the electron and hole densities of the considered transition for all compounds. The interpretation remains mostly the same as given by orbital analysis, except for compound **1c** that shows a highly local π–π* excitation on the central ring of the terpyridyl unit.

Fig. 5 The electron (green) and hole densities of the excitation of interest deduced from the one-particle transition density matrix. |

The first step in the analysis of charge separation upon electronic excitation is the identification of the state from which the fluorescence occurs. According to Liu et al.,^{12} the emitting state which controls the experimental quantum yield can be assigned to the specific π–π*/ICT state (Fig. 6). We consequently assume the π–π* state for compounds **1a**, **1b**, **1c** and **1d** and the ICT state for compounds **1e**, **1f**, and **1g**. All the incriminated transitions display large computed oscillator strengths (f > 1), except for compound **1c**, for which a very intense transition lies 0.4 eV above the considered state. For compounds **1a** and **1b**, LR-TDDFT overestimates the charge transfer excitation energy, giving rise to an ordering of the states different to that found using CC2 and ADC(2). For the other compounds, the ordering of the states is the same as the one obtained from CC2 and ADC(2) calculations. The excitation energies are, however, mostly overestimated (by more than 0.2 eV). Fig. 6 shows the orbitals participating in those transitions for compound **1a**. Pictures of orbitals for all remaining compounds are provided in the ESI† (Fig. S15–S21).

Fig. 6 Illustrative example of molecular orbitals involved in the two transitions of interest for compound 1a. The intense π–π* transition involves orbitals of the type of HOMO → LUMO whereas the intramolecular charge transfer has usually two main components, involving MO HOMO−1 → LUMO and from HOMO → LUMO+1. The details of all excitations of interest including orbitals can be found in the ESI† (Tables S31–S44 and Fig. S15–S21). |

The calculated energies are compared with energies corresponding to the maxima of the absorption bands in Fig. 7 and collected in Table 2. The observed differences can be attributed not only to the accuracy of the used computational method, but also to other factors. Firstly, the vertical excitation energies and the maxima of the absorption bands are not the same quantity. Secondly, the calculations are made in a vacuum at ground-state DFT geometries without taking into account the vibronic structure of the solvated chromophore. Large discrepancies between the vertical excitation energies calculated in the gas phase and the maxima of the absorption bands in non-polar solvents can be attributed to the flaws of the used quantum mechanical method. In the case of LR-TDDFT, the differences between the experimental and calculated energies are widely scattered (red crosses in Fig. 7 represent the correlation coefficient: 0.79), and this despite the use of the CAM-B3LYP functional, which is known to describe reasonably well the charge transfer excitations. Unfortunately, this method does not satisfy the necessary conditions to be used for the evaluation of the charge separation magnitudes.

Fig. 7 The calculated vertical excitation energies (π–π* for all compounds and ICT for 1c, 1e, 1f, 1g) calculated using the three considered methods vs. experimental maxima of the absorption bands (taken from ref. 12): red crosses: LR-TDDFT, green crosses: CC2, blue stars: ADC(2). The data points at correspond to compound 1f. |

Compound | Character | CAM-B3LYP^{a} |
CC2^{a} |
ADC(2)^{a} |
Exp.^{b} |
---|---|---|---|---|---|

a This work.b Y. Liu et al., J. Lumin., 2015, 157, 249. | |||||

1a |
π–π* | 4.2903 | 4.3759 | 4.3607 | 4.2606 |

1b |
π–π* | 4.3207 | 4.4018 | 4.3864 | 4.3200 |

1c |
π–π* | 4.5011 | 4.3770 | 4.3771 | 4.2901 |

1d |
π–π* | 4.1471 | 4.1472 | 4.1285 | 4.1605 |

1e |
π–π* | 4.1793 | 3.8877 | 3.8649 | 3.7916 |

1e |
ICT | 4.3927 | 4.2275 | 4.2141 | 4.1887 |

1f |
π–π* | 4.2619 | 4.2691 | 4.2664 | 3.7916 |

1g |
ICT | 4.0193 | 3.7662 | 3.7494 | 3.4633 |

1g |
π–π* | 4.4753 | 4.3552 | 4.2923 | 4.3656 |

The vertical excitation energies calculated with CC2 and ADC(2) show a remarkable agreement with each other within 0.02 eV for most excitation energies (Fig. 7 and Table S2, ESI†). The agreement with the energies corresponding to the experimental maxima of the absorption bands is not perfect, but of much better quality than that obtained with LR-TDDFT/CAM-B3LYP (green and blue crosses in Fig. 7 represent correlation coefficients: 0.87 and 0.85 respectively). For one molecule – compound **1f** – neither CC2 nor ADC(2) yield satisfactory excitation energies. Without taking into account compound **1f**, the slopes of the linear regression lines in Fig. 7 are 0.75 for CC2 and 0.74 for ADC(2), respectively. The corresponding correlation coefficients are 0.97 and 0.96, respectively. The exceptional failure of both ADC(2) and CC2 to yield reliable excitation energy for compound **1f** indicates that other excited state properties can be expected to be less accurately described using these methods. For this reason, although we report the obtained values of |Δ| for compound **1f**, we do not use it in the discussion of the phenomenological model linking |Δ| with Φ_{F}.

This exception calls for a further attention as it can be the result of (a) experimental measurement of Φ_{F}, (b) unbalanced errors in wavefunction at ground- and excited states obtained by the selected method, (c) inadequate description of the system (thermal motions of the system, the effect of the solvent), (d) evaluation of |Δ| for the wrong excited state (not the one which is involved in fluorescence), and last but not least (d) approximations underlying eqn (8). Since it is relatively straightforward to estimate in a rough way the effect of the solvent in which the measurements are made (CH_{2}Cl_{2}), a polarizable continuum model (PCM) with the corresponding dielectric constant ε = 8.93 combined with LR-TDDFT was used for this purpose. Neither the indirect effect of the solvent on the geometry nor the direct effect of the solvent on the excitation energy affect the excitation energy significantly enough (at most 0.1 eV and 0.12 eV, respectively) to explain the discrepancy between the experimental and calculated excitation energy in the case of compound **1f**. Another factor which might explain the observed discrepancy could be the used geometry at which the excitation energy is calculated. The planarization of the molecule lowers the excitation energy from 4.26 eV to 4.02 eV, thus reducing the overestimation by half. The value of the vertical excitation energy nevertheless remains out of the correlation. Note that a perpendicular arrangement of the aromatic rings drastically increases the excitation energies (Tables S76–S79 and Fig. S30 and S31, ESI†).

Fig. 8 Schematic representation of vector of the changes in dipole moments Δ computed for substituted 4′-phenyl-2,2′:6′2′′-terpyridines 1a–1g. The length of the arrows representing Δ is proportional to the change in the dipole moment given in Table 2. Δ is oriented along the pseudo two-fold molecular axis. |

We are now in a position to plot the experimental values of the spectroscopic π-conjugation length A_{π} calculated using eqn (5) (Table 1, column 10) as a function of the computed change in dipole moments. Fig. 9, blue dataset, shows that there is roughly a linear correlation between |Δ| and A_{π}. As a second measure of the charge separation upon excitation, the exciton sizes d_{he} have also been plotted against A_{π} in Fig. 9 (green diamonds), which indeed confirms the existence of a linear correlation.

(9a) |

(9b) |

Fig. 9 Computed change in the dipole moment (in [e Å^{−1}]) and exciton size (in [Å]) as a function of the π-conjugation length A_{π} calculated from eqn (5). Assuming a 10% uncertainty for the measured Φ_{F}, the size of the resulting error bars on A_{π} is smaller than the size of the symbols. The error bars are thus not shown. The linear regression lines used data for all compounds except for 1f (data shown in red). |

The estimations of A_{π} with the help of either |Δ| (eqn (9a)) or d_{he} (eqn (9b)) finally show satisfying sigmoidal correlation with the quantum yields (Fig. 10) for all terpyridyl compounds – except **1f**. Clearly, |Δ| computed for **1f** lies far from the sigmoidal curve. The reported value of |Δ| was evaluated for the first (and also the brightest) excited state of compound **1f**. For higher excited states, |Δ| is even smaller (between 3.24 and 4.7 for S2–S4, ESI†), which results in even larger deviation. Finally, the rough estimation of the solvent effects using the PCM model on top of LR-TDDFT(CAM-B3LYP) calculations shows a negligible solvent effect on |Δ| (Tables S17–S30 and Fig. S8–S14 in the ESI†). Due to the fact that neither ADC(2) nor CC2 predict correctly excitation energies for this compound, we attribute this deviation to the quantum mechanical method used to evaluate |Δ| more than to assumptions underlying eqn (8).

Fig. 10 Plot of the experimental fluorescence quantum yield (Φ_{F}) as a function of: left computed changes in dipole moments |Δ| right charge–hole separation distance for substituted 4′-phenyl-2,2′:6′2′′-terpyridines 1a–1g. The dotted trace corresponds to the best fit obtained with eqn (9a) and (9b) respectively. |

The results obtained in the present work indicate that the use of the charge separation parameters derived from ab initio calculations as a criterion for designing ligands of desired fluorescence properties is a promising strategy for improving light-downshifting. Although the present work provides a clear physical interpretation of the experimental parameter A_{π}, and a strategy to quantify it using ab initio calculations in the case of non-centrosymmetric rigid chromophores, the applied computational protocol requires refinements in more general cases. For centrosymmetric molecules, such as the ones considered in ref. 1, or molecules where a geometry change upon excitation is expected to be large, most likely the protocol should involve the optimisation of the lowest excited state geometry. This might brake the symmetry of centrosymmetric molecules and/or allow reordering the states to satisfy Kasha's rule. Taking into account the solvent, especially if it results in reordering of states is also unavoidable. The results of the present work suggest that such refinements are worthwhile trying.

Finally, we would like to point out that the use of low-costs methods such as the ones based on Linear-Response Time-Dependent DFT, using even approximations for the exchange–correlation energy designed to handle charge-transfer excitations (CAM-B3LYP in our case), proved not to be sufficiently accurate as far as excitation energies are concerned. They were, therefore, not considered for the further analysis of the magnitude of the charge separation.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6cp04252j |

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